JunJasonLuo ,Ka-SingLau Lipschitzequivalenceofself-similarsetsandhyperbolicboundaries

Advances in Mathematics 235 (2013) 555–579

www.elsevier.com/locate/aim

Lipschitz equivalence of self-similar sets and hyperbolic boundaries

Jun Jason Luo

, Ka-Sing Lau

aDepartment of Mathematics, The Chinese University of Hong Kong, Hong Kong bDepartment of Mathematics, Shantou University, Shantou, Guangdong 515063, PR China

Received 14 May 2012; accepted 17 December 2012

Communicated by Kenneth Falconer

Abstract

Kaimanovich (2003) [9] introduced the concept ofaugmented tree on the symbolic space of a self- similar set. It is hyperbolic in the sense of Gromov, and it was shown by Lau and Wang (2009) [12] that under the open set condition, a self-similar set can be identified with the hyperbolic boundary of the tree. In the paper, we investigate in detail a class ofsimpleaugmented trees and the Lipschitz equivalence of such trees. The main purpose is to use this to study the Lipschitz equivalence problem of the totally disconnected self-similar sets which has been undergoing some extensive development recently.

c

⃝2012 Elsevier Inc. All rights reserved.

Keywords: Augmented tree; Hyperbolic boundary; Incidence matrix; Lipschitz equivalence; OSC; Primitive;

Rearrangeable; Self-similar set; Self-affine set

1. Introduction

Two compact metric spaces(X,dX)and(Y,dY)are said to beLipschitz equivalent, and denote by X ≃Y, if there is a bi-Lipschitz mapσ fromX ontoY, i.e.,σ is a bijection and there is a constantC>0 such that

C⁻¹d_X(x,y)≤d_Y(σ (x), σ (y))≤C d_X(x,y) ∀x,y∈X.

∗Corresponding author at: Department of Mathematics, The Chinese University of Hong Kong, Hong Kong.

E-mail addresses:luojun2011@yahoo.com.cn(J.J. Luo),kslau@math.cuhk.edu.hk(K.-S. Lau).

0001-8708/$ - see front matter c⃝2012 Elsevier Inc. All rights reserved.

doi:10.1016/j.aim.2012.12.010

It is easy to see that if X ≃Y, then dim_H X = dim_HY = s, where dim_H denotes the Haus- dorff dimension. A more intensive study of this was due to Cooper and Pignartaro [2] in the late 80s, they showed that for certain Cantor sets X,Y onR, the Lipschitz equivalence implies that there exists a bi-Lipschitz mapσ and aλ >0 such thatH^s(σ (E))=λH^s(E). In another con- sideration, Falconer and Marsh [5] proved that for quasi-self-similar circles, they are Lipschitz equivalent if and only if they have the same Hausdorff dimension.

The recent interest of the Lipschitz equivalence is due to the path breaking study of Rao, Ruan and Xi [18] on a question of David and Semmes [3] on certain special self-similar set on Rto be dust-like (seeExample 5.2). They observed a graph directed relationship in the under- lying iterated function system (IFS), and made use of this to construct the needed bi-Lipschitz map. There is a number of generalizations onRandR^dfor the totally disconnected self-similar sets with uniform contraction ratio or with logarithmically commensurable contraction ratios ([4,19,23–25]). For certain Cantor-type sets onR^dwith non-equal contraction ratios, Rao, Ruan and Wang [17] had an elegant algebraic criterion for them to be Lipschitz equivalent, which im- proved a condition of Falconer and Marsh in [6]. Other considerations can be found in [15,16,22].

So far the investigation of the Lipschitz equivalence is very much restricted on a few special self-similar sets. In this paper, we will provide a broader framework to study the problem through the concept ofaugmented (rooted) tree. For an IFS {S_i}^m

i=1 of contractive similitudes onR^d (assume equal contraction ratio in the present situation) and the associated self-similar setK, we useX =∞

n=0Σⁿ,Σ = {1, . . . ,m}to denote the symbolic space. ThenX has a natural graph structure, and we denote the edge set byE_v(vfor vertical). We define a horizontal edge for a pair (u,v)inX×X ifu,v∈ΣⁿandSu(K)∩Sv(K)̸= ∅, and denote this set of edges byE_h. The augmented treeis defined as the graph(X,E)whereE=E_v∪E_h.

Such augmented tree was first introduced by Kaimanovich [9] on the Sierpinski gasket in order to incorporate the intersection of the cells to the symbolic space, and was developed by Lau and Wang [12] to general self-similar sets. It was proved that if an IFS satisfies the open set condition (OSC), then the augmented tree ishyperbolicin the sense of Gromov. There is a hyper- bolic metricρonX, which induces ahyperbolic boundary(∂X, ρ). The hyperbolic boundary is shown to be homeomorphic toK; moreover under certain mild condition, the homeomorphism is actually a H¨older equivalent map. This setup is used to study the random walks on(X,E)and their Martin boundaries [13].

Based on this, our approach to the Lipschitz equivalence of the self-similar sets is to lift the consideration to the augmented trees(X,E). We define ahorizontal connected componentof X to be the maximal connected horizontal subgraphT in some levelΣⁿ. Let C be the set of all horizontal connected components ofX. ForT ∈ C, we useTΣ to denote the set of offsprings ofT, and considerT ∪TΣ as a subgraph in X. We say thatT,T^′ ∈ C are equivalent if they are graph isomorphic. We callX simpleif there are finitely many equivalence classes. Under this condition, we can define an incidence matrix

A= [a_{i j}]

for the equivalence classes as follows: choose any componentT belonging to the classTi, and letZ_i1, . . . ,Z_iℓbe the connected components of the descendants ofT. The entrya_{i j}denotes the number ofZ_{i k}that belonging to the classTj.

It is shown that a simple augmented treeX is always hyperbolic, and the relationship of the hyperbolic boundary and the self-similar set is analogous to the case with OSC (Propositions 3.4 and3.5). Our basic theorem, to put it into a simple statement, is (assuming the IFS has equal contraction ratio):

Theorem 1.1. Suppose the augmented tree(X,E)is simple and the corresponding incidence matrix A is primitive (i.e., Aⁿ>0for some n>0), then∂(X,E)≃∂(X,E_v).

We call a self-similar setK dust-likeif it satisfiesS_i(K)∩S_j(K)= ∅fori̸= j. By reducing the Lipschitz equivalence on the trees inTheorem 1.1to the self-similar sets (Proposition 3.5, Theorem 3.10), we have

Theorem 1.2. (i)If in addition to the condition in Theorem1.1, the IFS satisfies some mild condition (H) (see Section2), then K is Lipschitz equivalent to a dust-like self-similar set.

(ii)If K and K^′are as in(i)and the two IFS’s have the same number of similitudes and the same contraction ratio, then they are Lipschitz equivalent.

The proof ofTheorem 1.1depends on constructing anear-isometrybetween the augmented tree(X,E)and(X,E_v). The crux of the construction is to use a technique ofrearrangementof edges (Section4), which is based on an idea of Deng and He in [4]. Actually we prove a less restrictive form ofTheorem 1.1(Theorem 3.7) in terms ofrearrangeable matrices.Theorem 1.1 follows from another theorem (Theorem 3.8) that the primitive property implies rearrangability.

We will provide an easy way to check an augmented tree being simple (Lemma 5.1), which is more efficient to apply to various examples than the graph directed systems that were used in the previous studies [4,18,24].Theorem 1.2essentially covers all the known cases so far, it also covers some new classes of IFS’s that have overlaps and rotations (see Section5). Moreover the theory can be extended from the self-similar IFS to the self-affine IFS: it is easy to see that the notion of augmented tree and the results on such tree remain unchanged. For self-affine sets onR^d, we can still establish the Lipschitz equivalence, making use of a device in [8] which re- places the Euclidean distance by an ultra-metric adapted to the underlying self-affine system (see Theorem 3.14).

The paper is organized as follows. In Section2, we recall some well-known results on hy- perbolic graphs and set up the augmented tree. In Section3, we introduce the notion ofsimple augmented tree, and derive its basic properties.Theorem 1.1is stated there, andTheorem 1.2to- gether with other consequences is proved. The proof ofTheorem 1.1and the involved concept of rearrangement are given in Section4. In Section5, we provide several new examples to illustrate our results; some concluding remarks and open questions are given in Section6.

2. Preliminaries

LetXbe a countably infinite set, we say thatX is agraphif it is associated with a symmetric subsetE of(X ×X)\ {(x,x) : x ∈ X}; we callx ∈ X avertex,(x,y) ∈ E anedge, which is more conveniently denoted byx ∼y(intuitively,x,yare neighborhoods to each other). By a pathinXfromxtoy, we mean a finite sequencex =x0,x1, . . . ,xn = ysuch thatxi ∼xi+1, i =0, . . . ,n−1. We always assume that the graphX is connected, i.e., there is a path joining any two verticesx,y ∈ X. We call X atreeif the path between any two points is unique. We equip a graphXwith an integer-valued metricd(x,y), which is the minimum among the lengths of the paths fromxtoy; the corresponding geodesic path is denoted byπ(x,y)and its length by

|π(x,y)|(=d(x,y)). Leto∈ Xbe a fixed point inX and call it therootof the graph. We use|x| to denoted(o,x), and say thatxbelongs to then-th level of the graph ifd(o,x)=n.

The notion of hyperbolic graph was introduced by Gromov [7,21]. First we define the Gromov product of any two pointsx,y∈ Xby

|x∧y| = 1

2(|x| + |y| −d(x,y)).

We say thatXishyperbolic(with respect too) if there isδ >0 such that

|x∧y| ≥min{|x∧z|,|z∧y|} −δ ∀x,y,z∈X.

Note that this is equivalent to a more geometric characterization: there exists aδ^′such that for any three points inX, the geodesic triangle isδ^′-thin: any point on one side of the triangle has distance less thanδ^′to some point on one of the other two sides.

For a fixeda >0 witha^′ =exp(δa)−1<√

2−1, we define anultra-metricρa(·,·)onX by

ρa(x,y)=

exp(−a|x∧y|) ifx̸=y,

0 otherwise. (2.1)

Then

ρa(x,y)≤(1+a^′)max{ρa(x,z), ρa(z,y)} ∀x,y,z∈ X, which is equivalent to the path metric

θa(x,y)=inf

 _n



i=1

ρa(xi−1,xi):n≥1, x=x0,x1, . . . ,xn=y, xi ∈ X

 . Sinceθaandρadetermine the same topology as long asa^′ <√

2−1, we will useρato replace θafor simplicity.

Definition 2.1. The hyperbolic boundary ofX is defined as∂X = ˆX \X whereXˆ is the com- pletion ofX underρa.

The metricρa can be extended onto ∂X, and under which∂X is a compact set. It is often useful to identifyξ ∈∂X with ageodesic rayinX that converge toξ, i.e., an infinite pathπ[x₁, x2, . . .] such that x_i ∼ xi+1 and any finite segment of the path is a geodesic. It is known that two geodesic raysπ[x1,x2, . . .], π[y1,y2, . . .]represent the sameξ ∈ ∂X if and only if

|xn∧yn| → ∞asn→ ∞.

Our interest is on the following tree structure introduced by Kaimanovich which is used to study the self-similar sets ([9,12]). For a tree X with a rooto, we use E_v to denote the set of edges (v for vertical). We introduce additional edges on each level{x : d(o,x)=n}, n ∈ N as follows. Letx^−k denote thek-th ancestor ofx, the unique point in the(n−k)-th level that is joined by a unique path.

Definition 2.2. LetXbe a tree with a rooto. Let E_h ⊂(X×X)\ {(x,x): x∈ X}such that it is symmetric and satisfies:

(x,y)∈E_h⇒ |x| = |y| and either x⁻¹=y⁻¹or(x⁻¹,y⁻¹)∈E_h.

We call elements inE_h horizontal edges, and forE = E_v∪E_h,(X,E)is called anaugmented tree.

Following [12], we say that a pathπ(x,y)is ahorizontal geodesicif it is a geodesic and it consists of horizontal edges only. It is called acanonical geodesicif there existu, v∈ π(x,y) such that:

(i)π(x,y)=π(x,u)∪π(u, v)∪π(v,y)withπ(u, v)a horizontal path andπ(x,u), π(v,y) vertical paths;

(ii) for any geodesic pathπ^′(x,y), dist(o, π(x,y))≤dist(o, π^′(x,y)).

Note that condition (ii) is to require the horizontal part of the canonical geodesic to be on the highest level. The following basic theorem was proved in [12]:

Theorem 2.3. Let X be an augmented tree. Then

(i)Letπ(x,y)be a canonical geodesic, then|x∧y| =l−h/2, where l and h are the level and the length of the horizontal part of the geodesic.

(ii)X is hyperbolic if and only if there exists a constant k>0such that any horizontal part of a geodesic is bounded by k.

The premier application of the augmented trees is to use their hyperbolic boundaries to study the self-similar sets. Throughout the paper, we assume a self-similar set K is generated by an iterated function system (IFS){S_i}^m

i=1onR^dwhere

S_i(x)=r R_ix+d_i, i =1, . . . ,m (2.2)

with 0<r <1,Ri is an orthogonal matrix, anddi ∈R^d. It is well-known thatK satisfiesK =

m

i=1Si(K).LetΣ= {1, . . . ,m}and letX =∞

n=0Σⁿbe the symbolic space representing the IFS (by convention,Σ⁰= ∅, and we still denote it byo). Foru=i1· · ·in, we useSuto denote the compositionSi₁◦ · · · ◦Sin.

LetE_vbe the set of vertical edges corresponding to the nature tree structure onX withoas a root. In [12], a set of horizontal edgesE_his defined as

E_h= {(u,v): |u| = |v|,u̸=vandK_u∩K_v̸= ∅},

whereKu = Su(K). Let E = E_v∪Eh, then(X,E)is an augmented tree induced by the self- similar set.

If the IFS is strongly separated (i.e.,S_i(K)∩S_j(K)= ∅fori ̸= j), thenKis calleddust-like.

It is clear that in this case,E_h = ∅, andρacoincides with the natural metric on the symbolic space:

ϱ(x,y)=exp(−amax{k:xi =yi, i ≤k}).

In [12], it was proved that under theopen set condition(OSC),Theorem 2.3(ii) implies that the above augmented tree is hyperbolic, and the nature mapΦ : ∂X → K is a homeomorphism.

Moreover if in addition, the IFS satisfies

condition (H): there exists a constantc>0 such that for any integern ≥1 and wordsu,v∈Σⁿ, K_u∩K_v= ∅ ⇒dist(K_u,K_v)≥crⁿ. (2.3) Then forα= −logr/a,Φsatisfies the following H¨older equivalent property:

C⁻¹|Φ(ξ)−Φ(η)| ≤ρa(ξ, η)^α ≤C|Φ(ξ)−Φ(η)| ∀ξ, η∈∂X.

Condition (H) is satisfied by the standard self-similar sets, for example, the generating IFS has the OSC and all the parameters of the similitudes are integers. However there are also examples that condition (H) is not satisfied (see [12] for an example such that the similitudes involve irrational translations).

FromDefinition 2.2, we see that the choice of the horizontal edges for the augmented tree can be quite flexible, for example we can useKu∩Kvto have positive dimension to defineE_h. In this paper, we will use another setting by replacingK with a bounded closed invariant setJ (i.e.,S_i(J)⊂Jfor eachi), namely

Eh= {(u,v): |u| = |v|,u̸=vandJu∩Jv̸= ∅}. (2.4)

We can take J = K as before or in many situations, take J = U for theU in the OSC (see the examples in Section5). The above statements on the hyperbolicity of the augmented tree and the homeomorphism of the boundary still valid by adopting the same proof; for the H¨older equivalence, we use the following modification of condition (H) forJ, which will be used again in provingProposition 3.5.

Lemma 2.4. Suppose the IFS in (2.2) satisfies condition (H), then for any bounded closed invariant set J , there exists c^′>0and k≥0such that for any n≥0andu,v∈Σⁿ,

Ju∩Jv= ∅ ⇒dist(Jui,Jvj)≥c^′rⁿ ∀i, j∈Σ^k.

Proof. Letcbe the constant in the definition of condition (H). For the bounded closed invariant setJ, we haveK ⊆Jand the Hausdorff distanced_H(K_i,J_i)≤c₁r^kfor alli∈Σ^k. In particular we chooseksuch thatc₁r^k<c/3.

Now ifu,v ∈ Σⁿ, Ju∩ Jv = ∅, then Ku∩Kv = ∅, it follows from condition (H) that dist(Ku,Kv)≥crⁿforu,v∈Σⁿ. Applying this and the above ton+k, we have

dist(J_ui,J_vj)≥dist(K_ui,K_vj)−d_H(K_ui,J_ui)−d_H(K_vj,J_vj)

≥crⁿ−(2c/3)rⁿ≥(c/3)rⁿ ∀i, j∈Σ^k. The lemma follows by takingc^′=c/3.

We remark that the augmented tree(X,E)depends on the choice of the bounded invariant set J. But under the OSC, the hyperbolic boundary is the same as they can be identified with the underlying self-similar set.

We conclude this section with the following simple relationship of the totally disconnected self-similar set and the structure of the augmented tree. The more explicit study of their Lipschitz equivalence will be carried out in detail in the rest of the paper. By a horizontal connected componentof an augmented tree, we mean a maximal connected horizontal subgraph on some levelΣⁿofX.

Proposition 2.5.Suppose the cardinality of any horizontal component in the augmented tree induced by the IFS in(2.2)is uniformly bounded, then the associated self-similar set K is totally disconnected.

The converse is also true if the about IFS is onR¹and satisfies the OSC.

Proof. Suppose K is not totally disconnected, then there is a connected component C ⊂ K contains more that one point. Note that for anyn > 0, K = 

i∈Σⁿ Ki. Let Ki₁ ∩C ̸= ∅. If C\Ki₁ ̸= ∅, then it is a relatively open set inC, and asCis connected,

∂C(C\K_i1)∩∂C(K_i1∩C)̸= ∅.

(∂C(E)means the relative boundary ofEinC). Letxbe in the intersection, there existsi₂∈Σⁿ such thatx∈ K_i1∩K_i2 andK_i2 ∩(C\K_i1)̸= ∅.

Inductively, ifk

j=1K_ij does not coverC, then we can repeat the same procedure to find i_k+1∈Σⁿsuch that

K_ik+1∩

 _k



j=1

K_ij



̸= ∅ and K_ik+1∩

 C\

k



j=1

K_ij



̸= ∅.

SinceK =

i∈ΣⁿKi, this process must end at some step, sayℓ, and in this caseC⊂_ℓ

j=1Ki_j. Since the diameter|Ki_j| =rⁿ|K| →0 asn → ∞,ℓmust tend to infinity, which contradicts the uniform boundedness of the horizontal connected componentsΣⁿ.

For the converse, we assume that the IFS is defined onR. Note that if K ⊂Ris totally dis- connected, then dim_H K =s<1 [20]. Letµdenote the restriction of thes-Hausdorff measure onK. Without loss of generality, we assumeµ(K)=1. Then it is well-known that for any point x ∈K and any 0<t<|K|(where|K|denotes the diameter ofK),

C₁<µ(B(x,t)) t^s <C₂,

whereC1,C2are constants independent ofxandt.

Supposei₁,i₂, . . . ,i_k is a finite sequence of distinct words inΣⁿ and is in a horizontal con- nected component (we take J =K for convenience), i.e.,K_ij ∩K_ij+1 ̸= ∅for 1≤ j ≤k−1.

LetGbe the smallest interval containing K_i1, . . . ,K_ik. Letx ∈ G∩K, and lett = |G|. Then the Hausdorff measureµimplies

µ(B(x,t))≥kr^ns.

This together witht = |G| ≤krⁿ|K|implies k^1−s

|K|^s = kr^ns

(krⁿ|K|)^s ≤ µ(B(x,t)) t^s ≤C2. Hencekis uniformly bounded.

3. Lipschitz equivalence and the main theorems

In this rest of the paper, unless otherwise stated, we will assume the augmented tree(X,E) is associated with the symbolic space of the IFS{S_i}^m

i=1in(2.2), andE =E_h∪E_vwhereE_his defined by a fixed bounded closed invariant set Jas in(2.4). We introduce a class of mappings between two hyperbolic graphs which plays a key role in the Lipschitz equivalence.

Definition 3.1. Let X andY be two hyperbolic graphs and letσ : X → Y be a bijective map.

We say thatσ is anear-isometryif there existsc>0 such that



|π(σ(x), σ (y))| − |π(x,y)|

≤c ∀x,y∈X.

Remark. By checkingπ(o,x), it is easy to show that the above definition implies

|σ(x)| −

|x|

≤c+3|σ (o)| +kwherekis the bound of the horizontal geodesic inTheorem 2.3(ii). Hence the above definition is equivalent to

|σ(x)| − |x|

<c, 

|π(σ (x), σ (y))| − |π(x,y)|

≤c ∀x,y∈ X (with different constantc).

Proposition 3.2. Let X , Y be two hyperbolic augmented trees that are equipped with the hyper- bolic metrics with the same parameter a (as in(2.1)). Suppose there exists a near-isometryσ : X →Y , then ∂X ≃∂Y .

Proof. With the notation as inTheorem 2.3(i), it follows that forx̸=y∈X,

|π(x,y)| = |x| + |y| −2l+h, |π(σ (x), σ(y))| = |σ(x)| + |σ (y)| −2l^′+h^′.

From the definition ofσ (and the remark), we have



|σ(x)| − |x|

, 

|σ (y)| − |y|

≤c, |l−l^′| ≤3c/2+k/2, and |h−h^′| ≤k for somek>0 (wherekis the hyperbolic constant as inTheorem 2.3(ii)). ByTheorem 2.3(i),

|x∧y| =l−h/2 and |σ (x)∧σ (y)| =l^′−h^′/2. It follows that



|σ(x)∧σ(y)| − |x∧y|

= |l^′−h^′/2−l+h/2| ≤3c/2+k:=k^′.

Letλ=e^ak′, together with the definition of the ultra-metricρa(x,y)=exp(−a|x∧y|)in(2.1), we conclude that

λ⁻¹ρa(x,y)≤ρa(σ (x), σ (y))≤λρa(x,y) ∀x,y∈ X.

By extending the metrics to the boundaries∂X, ∂Y, the above impliesσ is a bi-Lipschitz map from∂Xonto∂Y.

LetC be the set of all horizontal connected components ofX. ForT ∈C, we letTΣ = {ui: u∈T,i ∈Σ}be the set of offsprings ofT. Note that if two distinct componentsT,T^′ ∈C lie in the same level, thenTΣis not connected toT^′Σ, equivalently,





i∈TΣ

Ji



∩







j∈T^′Σ

Jj



= ∅. (3.1)

This follows easily fromS_ui(J)∩S_vj(J)⊂S_u(J)∩S_v(J)= ∅for allu∈T, v∈T^′,i,j ∈Σ. By regardingT ∪TΣ as a subgraph inX. We say thatT,T^′∈C areequivalent, denoted by T ∼T^′, if there exists a graph isomorphism

g: T ∪TΣ → T^′∪T^′Σ,

that is,g is a bijection such thatgandg⁻¹preserve the vertical and horizontal edges. It is easy to check that∼is indeed an equivalence relation. We use[T]to denote the equivalence class and call it aconnected classdetermined byT. Obviously,{o}is the connected class determined by the rooto.

Definition 3.3. An augmented tree X is calledsimple (with respect to the defining bounded closed invariant setJ) if there are finitely many connected classes, i.e.,C/∼is finite.

Proposition 3.4.A simple augmented tree is always hyperbolic.

Proof. Note that for each geodesicπ(x,y)inX, the horizontal part must be contained in a hori- zontal component of the augmented tree. Since there are finitely many connected classes[T], and eachT contains finitely many vertices, it follows that the horizontal part ofπ(x,y)is uniformly bounded, and hence hyperbolic byTheorem 2.3(ii).

In the following we show that the hyperbolic boundary of a simple augmented tree is H¨older equivalent to the self-similar set, which is similar to the case in [12] for the OSC.

Proposition 3.5.Let {S_i}^m

i=1 be an IFS satisfies condition (H) in (2.3), and assume that the corresponding augmented tree(X,E)is simple. Then there exists a bijectionΦ : ∂X → K

satisfying the H¨older equivalent property:

C⁻¹|Φ(ξ)−Φ(η)| ≤ρa(ξ, η)^α ≤C|Φ(ξ)−Φ(η)|, (3.2)

whereα= −logr/a and C >0is a constant.

Proof. The proof is essentially the same as in [12] by replacingKwith the invariant set JinE_h. We sketch the main idea of proof here. For any geodesic rayξ =π[u₁,u₂, . . .], we define

Φ(ξ)= lim

n→∞S_un(x₀)

for somex0∈ J. Then the mapping is well-defined and is a bijection (see Lemma 4.1, Theorem 4.3 in [12]).

To show that Φ satisfies (3.2), let ξ = π[u₀,u₁,u₂, . . .], η = π[v₀,v₁,v₂, . . .]be any two non-equivalent geodesic rays inX. Then there is a canonical bilateral geodesicγ joiningξ andη:

γ =π[. . . ,u_n+1,u_n,t₁, . . . ,t_ℓ,v_n,v_n+1, . . .] withu_n,t₁, . . . ,t_ℓ,v_n∈Σⁿ. It follows that

|S_un(x₀)−S_vn(x₀)| ≤(ℓ+2)rⁿ|J|.

SinceX is simple,ℓis uniformly bounded (byProposition 3.4). Note thatΦ(ξ)∈ J_uk andΦ(η)

∈ J_vk for allk≥0, hence

|Φ(ξ)−Su_n(x₀)|, |Φ(η)−Sv_n(x₀)| ≤rⁿ|J|. Therefore

|Φ(ξ)−Φ(η)| ≤ |Φ(ξ)−Su_n(x0)| + |Su_n(x0)−Sv_n(x0)| + |Φ(η)−Sv_n(x0)|

≤ C₁rⁿ.

Sinceγ is a bilateral canonical geodesic, we have|ξ∧η| =n−(ℓ+1)/2 andℓis uniformly bounded. By usingρa(ξ, η)=exp(−a|ξ∧η|), we see that

|Φ(ξ)−Φ(η)| ≤Cρa(ξ, η)^α.

On the other hand, assume thatξ ̸=η. Sinceγis a geodesic, it follows that(un+1,vn+1)̸∈E_h, and henceJu_n+1∩Jv_n+1 = ∅. ByLemma 2.4, there isk(independent ofn) such that

Ju∩Jv= ∅ ⇒dist(Jui,Jvj)≥c^′rⁿ ∀i, j∈Σ^k.

Referring toγ =π[. . . ,un+1,u_n,t1, . . . ,t_ℓ,v_n,vn+1, . . .], we haveΦ(ξ) ∈ Ju_n+k+1, Φ(ξ)∈ Jv_n+k+1. It follows that

|Φ(ξ)−Φ(η)| ≥dist(Ju_n+k+1,Jv_n+k+1)≥c^′rⁿ,

and|Φ(ξ)−Φ(η)| ≥c^′′ρa(ξ, η)^α follows by the definition ofρaas the above.

For a simple augmented treeX, we label the connected classes as{T1, . . . ,Tr}and introduce anr×rincidence matrix

A= [ai j]_r×r (3.3)

for the connected classes. The entries a_{i j} are defined as follows. For any 1 ≤ i ≤ r, take a horizontal connected componentT inXsuch that[T] =Ti. LetZi1, . . . ,Z_iℓbe the horizontal connected components consisting of offsprings generated byT, i.e.,TΣ =_ℓ

k=1Zi k, and define a_{i j} =#{k:1≤k≤ℓ,[Z_{i k}] =Tj}.

Observe thata_{i j} is independent of the choice of the components in the equivalence classes. It is clear that forT,T^′∈C,[T] = [T^′]implies #T =#T^′. But the converse is not true. As a direct consequence of the definition, we have

Proposition 3.6.Letb= [b₁, . . . ,b_r]^t where b_i =#T where[T] =Ti, then Ab=mb.

The following theorem is for Lipschitz equivalence on the hyperbolic boundaries, it is the crucial step to establish the equivalence for the self-similar sets.

Theorem 3.7. Suppose the augmented tree(X,E)is simple, and suppose the corresponding in- cidence matrix A is(m,b)-rearrangeable (where the m andbare defined as inProposition3.6).

Then there is a near-isometry between(X,E)and(X,E_v), so that∂(X,E)≃∂(X,E_v).

The notion ofrearrangeable matrixis an important tool to construct the near-isometry. Since the concept is a little complicated, we will introduce this in more detail, and proveTheorem 3.7 together with the following theorem in the next section.

Theorem 3.8. If the incidence matrix A is primitive, then A^kis(m^k,b)-rearrangeable for some k>0. Consequently ∂(X,E)≃∂(X,E_v).

As a direct consequence, we have

Corollary 3.9. Under the assumption onTheorem3.7(orTheorem3.8), then(∂(X,E), ρa)is totally disconnected.

ByTheorem 3.7we obtain the following Lipschitz equivalence on the self-similar sets.

Theorem 3.10. Let K and K^′be self-similar sets that are generated by two IFS’s as in(2.2)with the same number of similitudes and the same contraction ratio, and satisfy condition (H) in(2.3).

Assume the associated augmented trees are simple and the incidence matrices are (m,b)- rearrangeable (in particular, primitive). Then K and K^′are Lipschitz equivalent, and are also Lipschitz equivalent to a dust-like self-similar set.

Proof. It follows fromTheorem 3.7that

∂(X,E)≃∂(X,E_v)=∂(Y,E_v)≃∂(Y,E) (3.4) (for the respective metricsρa). Letϕ : ∂(X,E) → ∂(Y,E)be the bi-Lipschitz map. With no confusion, we just denote these two boundaries by∂X,∂Y as before.

ByProposition 3.5, there exist two bijectionsΦ₁ :∂X → K andΦ₂: ∂Y → K^′satisfying (3.2)with constantsC1,C2, respectively. Defineτ :K →K^′as

τ =Φ₂◦ϕ◦Φ₁⁻¹. Then

|τ(x)−τ(y)| ≤C₂ρa(ϕ◦Φ₁⁻¹(x), ϕ◦Φ₁⁻¹(y))^α

≤ C₂C₀^αρa(Φ₁⁻¹(x),Φ₁⁻¹(y))^α

≤ C2C₀^αC1|x−y|.

LetC^′=C2C₀^αC1, then

|τ(x)−τ(y)| ≤C^′|x−y|.

Similarly, we haveC^′−1|x−y| ≤ |τ(x)−τ(y)|. Thereforeτ :K → K^′is a bi-Lipschitz map.

For the last statement, we can regard(X,E_v)as the augmented tree of an IFS that is strongly separated, and then apply the above conclusion.

Corollary 3.11. The IFS inTheorem3.10satisfies the OSC.

Proof. We make use of the following well-known result of Schief [20] on a self-similar setK: let sbe the similarity dimension ofK, then the IFS satisfies the OSC if and only if 0<H^s(K) <∞. LetK be the self-similar set as inTheorem 3.10, then it is Lipschitz equivalent to a dust-like set K^′′. It follows that 0 < H^s(K^′′) < ∞, so is K by the Lipschitz equivalence. Hence by Schief’s criterion, the IFS forK satisfies the OSC.

In Section5, we will provide some interesting examples for the Lipschitz equivalence of the totally disconnected self-similar sets inTheorem 3.10. We also remark that inTheorem 3.10the condition on the augmented tree can be weaken, and the proof still yields a very useful result.

Proposition 3.12. Let K and K^′ be self-similar sets that are generated by two IFS’s as in (2.2)that have the same number of similitudes, same contraction ratio, and satisfy condition (H) in(2.3). Suppose the two IFS’s satisfy either(i)the OSC, or (ii)the augmented trees are simple. Then

K ≃K^′⇔∂X≃∂Y. (3.5)

Proof. The sufficiency of(3.5)is always satisfied, as we can replace(3.4)by the given condition

∂X ≃∂Y, then follows from the same proof ofTheorem 3.10. The necessity follows by making use of the H¨older equivalence(3.2)which is satisfied for cases (i) and (ii), and proceeds with a similar estimation forϕ =Φ₂⁻¹◦τ◦Φ₁.

We remark that the above theory of Lipschitz equivalence can also be applied to study the self-affine systems. LetBbe ad×dexpanding matrix (i.e., all the eigenvalues have moduli>1) and let{Si}^m

i=1withSi(x)=B⁻¹(x+di),di ∈R^dbe the IFS. For the part of simple augmented tree, it is clear that the notion can be defined and the hyperbolicity inProposition 3.4follows by the same way. Moreover we have

Proposition 3.13. For the IFS {S_i}^m

i=1 of self-affine maps as the above, Theorems 3.7 and 3.8remain valid.

For the part involves the self-affine set onR^d, we need to use a device in [8] by replacing the Euclidean norm with an “ultra-norm” adapted to the matrix B. By renorming, we can assume without loss of generality that∥x∥ ≤ ∥B x∥. For 0< δ <1/2, letϕ≥0 be aC^∞function sup- ported in the open ballU_δcentering at 0 withϕ(x)=ϕ(−x)and

R^d ϕ=1. LetV =BU1\U1, and leth=χV ∗ϕbe the convolution of the indicator functionχV andϕ. Letq = |det(B)|and define

w(x)=

∞



n=−∞

q^−n/qh(Bⁿx) x∈R^d.

Thenw(x)satisfies (i) w(x) = w(−x) andw(x) = 0 if and only if x = 0, (ii)w(B x) = q^1/dw(x), and (iii) there existsβ >1 such thatw(x+y)≤βmax{w(x), w(y)}. Thiswis used as a distance (ultra-metric) to replace the Euclidean distance to define the generalized Hausdorff measureH^α_w, Hausdorff dimension dim^w_H, box dimension dim^w_B. Under this setting, most of the basic properties for the self-similar sets (including Schief’s basic result on OSC) can be carried to the self-affine sets [8]. To apply to here, we need to adjust condition (H)(2.3)to

K_u∩K_v= ∅ ⇒dist_w(K_u,K_v)≥cq^−n/d

and to replace therⁿin the proofs ofLemma 2.4andProposition 3.5byq^−n/d. Then we have Theorem 3.14. With K and K^′ self-affine sets satisfying the conditions inTheorem3.10. The K and K^′ are Lipschitz equivalent under the ultra-metric defined byw, and they are Lipschitz equivalent to a dust-like self-affine set.

4. Rearrangeable matrix and proofs of the main theorems

The proof of the Lipschitz equivalence of the simple augmented tree to the original tree in Theorem 3.7is to construct a near-isometry between them, which is based on a device of “rear- rangement” of graphs. The idea of rearrangement was introduced by Deng and He [4]. A similar technique of “equal decomposition” was also used to consider the Lipschitz equivalence in [18]

(see also [24,25]). First we give a detail discussion of the concept of rearrangement.

Definition 4.1. Given m,r ∈ N. Suppose a = [a₁, . . . ,a_r] ∈ Z^r+ is a row vector, and b= [b₁, . . . ,b_r]^t ∈N^r is a column vector. We say thatais(m,b)-rearrangeable if there exists an integerp>0, and a nonnegative integral matrixC= [c_{i j}]_p×r such that

a= [1, . . . ,1]C and Cb= [m, . . . ,m]^t. (Note that in this caseab=pmfor somep∈N.)

A matrixAis called(m,b)-rearrangeable if each row of Ais(m,b)-rearrangeable.

Remarks. (1) The intuitive explanation of the definition is as follows. Leta_i be the number of balls with the same weightb_i. Thatais(m,b)-rearrangeable means we can rearrange these balls intopgroups (thep-rows inC) such that in each group the number of balls with weightb_j isc_{i j} and the total weight is exactlym. It is clear that the total weight of all balls is pm=ab=1Cb.

(2) It follows easily from the definition that ifais(m,b)-rearrangeable, then max{b_i :a_ib_i ̸=

0} ≤m.

(3) If we write anr×rmatrixAas

a₁ ... ar



. ThenAis(m,b)-rearrangeable is equivalent to the existence ofp= [p1, . . . ,pr]^t ∈N^r such thatAb=mpand a sequence of nonnegative integral matrices{C_i}^r

i=1such thata_i =1C_i andC_ib= [m, . . . ,m]^t ∈N^pi for alli.

We will prove some sufficient conditions for(m,b)-rearrangeable in the sequel (Lemma 4.7, Proposition 4.8). In the following we use two examples to illustrate more on such notion.

Example 4.2.Let a ∈ Z^r+, and letb = 1^t. Suppose

iai = m, then trivially, a is(m,b)- rearrangeable (with p = 1, C = a, i.e., intuitively we put everything in one group). Con- sequently, for any nonnegative integral matrix Awithm as an eigenvalue and b = 1^t as the eigenvector, it is(m,b)-rearrangeable.

Example 4.3. Let A=





1 1 0

1 1 1

1 1 2





andb= [1,2,3]^t. ThenAb=3b, andAis(3,b)-rearrangeable, so isA².

Proof. To show that Ais(3,b)-rearrangeable, it suffices to check on each row of Ais(3,b)- rearrangeable:

fora= [1,1,0], thenab=3, we takeC=a;

fora= [1,1,1], thenab=2×3, we takeC=_{1 1 0}

0 0 1



; fora= [1,1,2], thenab=3×3, we takeC=

_{1 1 0}

0 0 1

0 0 1

 . For A² =

_{2 2 1}

3 3 3

4 4 5



, we can proceed in the same way to show that A² is also (3,b)- rearrangeable. The corresponding matricesCfor the three rows of A²are the transposes of the following matrices:

_{1 1 0}

1 1 0

0 0 1



,_{1 1 0 1 0 0}

1 1 0 1 0 0

0 0 1 0 1 1

 and

_{1 1 0 1 0 0 1 0 0}

1 1 0 1 0 0 1 0 0

0 0 1 0 1 1 0 1 1



.

As a matter of fact, the above example is typical by the following proposition.

Proposition 4.4. Letb= [b1, . . . ,b_r]^t ∈N^r. If a matrix A= [a_{i j}]_r×r is(m,b)-rearrangeable, then Aⁿis(m,b)-rearrangeable for n≥1.

If in addition, the eigen-relation Ab=mbis satisfied, then Aⁿis also(mⁿ,b)-rearrangeable.

Proof. We use induction to prove the first part. Leta_i, 1 ≤ i ≤ r be the row vectors of A.

Since Ais(m,b)-rearrangeable, there existp = [p₁, . . . ,p_r]^t ∈ N^r such thatAb=mpand a sequence of matrices{C_i}^r

i=1such thata_i =1C_i andC_ib= [m, . . . ,m]^t ∈N^pi for alli. Assume thatAⁿ⁻¹is(m,b)-rearrangeable, thenAⁿ⁻¹b=mpfor some positive integer vector

p. ConsiderAⁿ, letαi be thei-th row ofAⁿ. SinceAⁿb=m Ap, we haveαib=ma_ip:=mqi. WriteAⁿ⁻¹= [

ai j], it follows that αi =

r



j=1

ai ja_j =

r



j=1

ai j1C_j. Let

C⁽ⁱ⁾=

C₁, . . . ,C₁

  

a_i1

, . . . ,C_r, . . . ,C_r

  

a_ir

t

where the transpose means transposing the row of matrices into a column of matrices (without transposing theC_j itself). Then

αi =1C⁽ⁱ⁾ and C⁽ⁱ⁾b= [m, . . . ,m]^t ∈N^qi.

Henceαi is(m,b)-rearrangeable, and Aⁿis(m,b)-rearrangeable.

For the second part, since Ab = mband Aⁿb =mⁿb, we can replace the previous integral vectorpbyb. Thenαib =mqi =mⁿpi. We then replace theqi ×r matrixC⁽ⁱ⁾ in the above

by thep_i×rmatrixD⁽ⁱ⁾which is obtained by summing every consecutivemⁿ⁻¹row vectors of C⁽ⁱ⁾(note thatq_i =mⁿ⁻¹p_i). Hence

αi =1D⁽ⁱ⁾ and D⁽ⁱ⁾b= [mⁿ, . . . ,mⁿ]^t ∈N^pi

so thatαi is(mⁿ,b)-rearrangeable. Consequently,Aⁿis(mⁿ,b)-rearrangeable.

Proof of Theorem 3.7. Recall that Ab = mb whereb = [b1, . . . ,br]^t withbi = #T where [T] =Ti (Proposition 3.6). First we claim that we can assume without loss of generality that max_ibi ≤ m. For otherwise, let k be sufficiently large such that max_ibi ≤ m^k, byProposi- tion 4.4,A^kis(m^k,b)-rearrangeable. The IFS of thek-th iteration of{S_i}^m

i=1has symbolic space X^′=∞

n=0Σ^knand the augmented tree has incidence matrixA^k; moreover the two hyperbolic boundaries∂X^′ and∂X are identical. Hence we can consider A^k instead if max_ib_i ≤ mis not satisfied.

Let X₁ =(X,E), X₂ = (X,E_v). In view ofProposition 3.2, it suffices to show that there exists a near-isometryσ betweenX₁andX₂, and hence∂(X,E)≃∂(X,E_v). We define thisσ to be a one-to-one mapping fromΣⁿ(inX₁) toΣⁿ(inX₂) inductively as follows: Let

σ(o)=o and σ (i)=i, i ∈Σ.

Supposeσis defined on the levelnsuch that for every horizontal connected componentT,σ(T) has the same parent (seeFig. 1), i.e.,

σ(x)⁻¹=σ(y)⁻¹ ∀x,y∈T ⊂Σⁿ. (4.1)

To define the mapσ onΣⁿ⁺¹, we note thatT inΣⁿgives rise to horizontal connected compo- nents inΣⁿ⁺¹, which are accounted by the incidence matrixA. We can write

TΣ =

ℓ

k=1

Z_k

whereZkare horizontal connected components consisting of offsprings ofT. IfT belongs to the connected classTi, then #T =bi. By the definition of the incidence matrix AandAb =mb, we have

bim=

ℓ

k=1

#Zk =

r



j=1

ai jbj.

Since A is (m,b)-rearrangeable, for the a_i, there exists a nonnegative integral matrix C = [c_{s j}]_b

i×r (depends oni) such that

a_i =1C and Cb= [m, . . . ,m]^t.

We decomposea_i into bi groups according toC as follows. Note thatai j denotes the number of Zk that belongs toTj. For each 1 ≤ s ≤ bi and 1 ≤ j ≤ r, we choosecs j of those Zk, and denote byΛ_s the set for all the chosenkwith 1≤ j ≤ r. Then we can write the index set {1,2, . . . , ℓ}as a disjoint union:

{1,2, . . . , ℓ} =

bi



s=1

Λ_s.

Fig. 1. An illustration of the rearrangement byσ, the•,◦and×on the left denote the types of connected components.

Hence_ℓ

k=1Zkcan be rearranged asbi groups so that the total size of every group is equal to m, namely,

ℓ

k=1

Z_k= 

k∈Λ1

Z_k∪ · · · ∪ 

k∈Λ_bi

Z_k. (4.2)

Note that each set on the right hasmelements.

For the connected componentT = {i₁, . . . ,i_bi} ⊂Σⁿ, we have definedσ onΣⁿandσ(T)

= {j1 =σ (i1), . . . ,j_bi =σ (i_bi)}.In view of(4.2), we defineσ onTΣ =_ℓ

k=1Z_kby assign- ing each

k∈Λs Zk (it hasmelements) themdescendants ofjs (seeFig. 1). It is clear thatσ is well-defined onTΣ and satisfies(4.1)forx,y∈ TΣ. We apply the same construction ofσ on the offsprings of every horizontal connected component inΣⁿ. It follows thatσ is well-defined and satisfies(4.1)onΣⁿ⁺¹. Inductively,σ can be defined fromX1toX2and is bijective.

Finally we show thatσis indeed a near-isometry and complete the proof. Sinceσ : X1→X2

preserves the levels, without loss of generality, it suffices to prove the near-isometry forx,ybe- long to the same level. Let π(x,y)be the canonical geodesic connecting them, which can be written as

π(x,y)= [x,u₁, . . . ,u_n,t₁, . . . ,t_k,v_n, . . . ,v₁,y]

where[t₁, . . . ,t_k]is the horizontal part and[x,u₁, . . . ,u_n,t₁], [t_k,v_n, . . . ,v₁,y]are vertical parts. Clearly,{t₁, . . . ,t_k}must be included in one horizontal connected component of X₁, we denote it byT_j. With the notation as inTheorem 2.3(i), it follows that forx̸=y∈X₁,

|π(x,y)| = |x| + |y| −2l+h, |π(σ (x), σ(y))| = |σ (x)| + |σ (y)| −2l^′+h^′. We have



|π(σ(x), σ(y))| − |π(x,y)|

≤ |h−h^′| +2|l^′−l| ≤k+2|l^′−l| wherekis a hyperbolic constant as inTheorem 2.3(ii). IfT_j is a singleton, then

|l^′−l| =0.

IfTj contains more than one point, then the elements ofσ (Tj)share the same parent. Then the confluence ofσ (x)andσ (y)(as a tree) isσ (x)⁻¹(=σ(y)⁻¹). Hence

|l^′−l| =1. Consequently

|π(σ(x), σ(y))| − |π(x,y)|

≤k+2.

This completes the proof thatσ is a near-isometry and the theorem is established.

Corollary 4.5. Under the same assumption of Theorem3.7. If there exists k∈Nsuch that A^kis (m^k,b)-rearrangeable, then∂(X,E)≃∂(X,E_v).

To prove ofTheorem 3.8that primitive implies(m,b)-rearrangeable, we need a combinatorial lemma due to Xi and Xiong [24]. We include their proof for completeness.

Lemma 4.6. Let p,m and {n_i}_i∈Λ be positive integers with 

i∈Λn_i = pm. Suppose there exists an integer l with n_i ≤l <m for all i ∈Λ, and#{i ∈ Λ:n_i =1} ≥ pl. Then there is a decompositionΛ=p

s=1Λ_s satisfying

i∈Λsn_i =m for1≤s≤ p.

Proof. The lemma is trivially true for p =1, suppose it is true for p. For p+1, letΩ ⊂ {i : n_i =1}with #Ω=(p+1)l, and select a maximal subset∆₁ofΛ\Ωsuch that

i∈∆1n_i <m.

We claim that



i∈∆₁

n_i ≥m−l.

For otherwise, takei₀ ∈ Λ\(∆₁∪Ω), then

i∈∆1∪{i0}n_i < (m−l)+l = m, which con- tradicts the maximality of ∆₁ and the claim follows. Choose a subset Ω₁ from Ω such that

#Ω₁=m−

i∈∆1n_i (≤l)and setΛ₁=∆₁∪Ω₁, then

i∈Λ1n_i =m.

Note that forΛ^′=Λ\Λ₁, #{i ∈Λ^′:n_i =1} ≥ pl. Applying the inductive hypothesis on p, we get a decompositionΛ^′ =p+1

s=2Λ_s with

i∈Λsni =mfors ≥2. Therefore, the assertion for p+1 holds, and the lemma is proved.

Lemma 4.6yields the following rearrangement lemma we need (see also [4]).

Lemma 4.7. Letb= [b1, . . . ,b_r]^t ∈N^r with b1=1. Letℓ=max_jb_j anda= [a1, . . . ,a_r] ∈ Z^r+ such that a1 ≥ ℓ². Suppose there exists m > ℓ such that ab = pm for some integer 0< p≤ℓ, thenais(m,b)-rearrangeable.

Proof. We first assume that alla_i >0. By the assumption, we haveℓ <manda₁≥ℓ² ≥ pℓ. Define a sequence{n_j}^r

j=1by

n_j =



















b1 j=1, . . . ,a1;

b₂ j=a₁+1, . . . , a₁+a₂; ...

b_r j=

r−1



j=1

a_j +1, . . . ,

r



j=1

a_j.

Note thatn_j ≤ℓ. LetΛ= {1,2, . . . , r

j=1a_j}be the index set. Then the assumptionab= pm is equivalent to:



j∈Λ

n_j = pm.

ByLemma 4.6, there is a decompositionΛ=p

s=1Λ_ssatisfying

j∈Λsn_j =mfor 1≤s≤ p.

Counting the number ofbj’s in each groups,

c_{s j} =#{k∈Λ_s :n_k =b_j}, 1≤s≤ p, 1≤ j≤r. The lemma follows by letting the matrixC= [c_{s j}]_p×r.

If some of the a_i equals zero. Without loss generality we assume that a_r = 0 and a_i >

0, 1 ≤ i ≤ r −1. Leta^′ = [a₁, . . . ,a_r−1]andb^′ = [b₁, . . . ,b_r−1], thena^′b^′ = pm and by the above conclusion,a^′is(m,b)-rearrangeable by a p×(r −1)matrixC^′. LetC be the p×r matrix obtained by adding a last column0toC^′. Then it is easy to see thatais(m,b)- rearrangeable.

Proposition 4.8. Let A be an r×r nonnegative integral matrix and is primitive (i.e., Aⁿ >0 for some n >0). Letb= [b1, . . . ,br]^t ∈N^r with b1=1and Ab=mbfor some m ∈N. Then

A^kis(m^k,b)-rearrangeable for some k>0.

Proof. Let ℓ = max_jb_j. Observe that A is primitive, there exists a large k such that ℓ <

m^k, and A^k > 0 with every entry greater thanℓ². Hence Lemma 4.7implies A^k is (m^k,b)- rearrangeable.

Proof of Theorem 3.8. Let Abe the incidence matrix, and Ab =mbwhereb= [b₁, . . . ,b_r]^t withb_i = #T where[T] = Ti (Proposition 3.6). If we let the roototo beT1, then it is clear from the definitions ofA,bandAb=mbthatb₁=1. HenceProposition 4.8implies thatA^kis (m^k,b)-rearrangeable for somek>0.

To prove the last part, we can assume without loss of generality thatA is(m,b)-rearrangeable andmax_ib_i ≤ m. For otherwise, by the primitive assumption on Aand by the same reasoning as in the proof ofTheorem 3.7, we can consider the augmented tree as the IFS defined by the k-th iteration of{S_i}^m

i=1, and the correspondingA^kis(m^k,b)-rearrangeable. Hence we can apply Theorem 3.7and complete the proof ofTheorem 3.8.

5. Examples

In this section, we provide several examples to illustrate our theorems of simple augmented tree and Lipschitz equivalence. Note that all the IFS’s considered here satisfy condition (H);

also the bounded closed invariant setsJ we use are connected tiles and they have finite number of neighbors. By using the following lemma, we see that the process of finding the connected components will end in finitely many steps.

We assume the IFS consists of contractive similitudesS_i(x)= B⁻¹(x+d_i),i =1, . . . ,m, whereB⁻¹=r Ris the scaled orthogonal matrix. LetJbe a closed subset such thatS_i(J)⊂J for each i. Then for i = i₁· · ·i_k ∈ Σ^k, we have J_i = S_i(J) = B^−k(J +d_i) where d_i=d_ik +Bd_ik−1 + · · · +B^k−1d_i1.

Lemma 5.1. For two horizontal connected components T₁⊂Σ^k1,T₂⊂Σ^k2 with#T₁=#T₂= ℓ. If there exist similitudesφi(x)=B^kix+c_i, i =1,2, where c_i ∈R^dsuch that

φ1(∪_i∈T

1 Ji)=φ2(∪_j∈T

2 Jj)=J∪(J+v1)∪ · · · ∪(J+vℓ−1) for some vectorsv1, . . . , v_ℓ−1∈R^d, then T₁∼T₂.

Proof. By the definition and(3.1), it suffices to proveT1ΣandT2Σhave the same connectedness structure, equivalently, to prove this for

i∈T₁

m

i=1Jiiand

j∈T₂

m

j=1Jjj. Lettingv0=0, we